Number
Standard Form
Year 10 · Year 11
- ✓By the end of this lesson students will be able to write very large and very small numbers in standard form.
- ✓By the end of this lesson students will be able to convert numbers from standard form back to ordinary numbers.
- ✓By the end of this lesson students will be able to order numbers written in standard form.
- ✓By the end of this lesson students will be able to perform calculations (multiplication, division, addition, and subtraction) with numbers in standard form.
Key concepts
Standard form (also known as scientific notation) is a concise way to write very large or very small numbers. A number is written in standard form when it is expressed as A × 10ⁿ, where 'A' is a number such that 1 ≤ A < 10, and 'n' is an integer (a whole number, positive, negative, or zero).
To write a number in standard form, you need to determine 'A' and 'n'. 1. Place the decimal point so that there is only one non-zero digit to its left. This gives you the value of 'A'. 2. Count how many places you moved the decimal point. This number is 'n'. * If you moved the decimal point to the left (for a large number), 'n' is positive. * If you moved the decimal point to the right (for a small number), 'n' is negative.
To convert a number from standard form (A × 10ⁿ) back to an ordinary number: 1. If 'n' is positive, move the decimal point 'n' places to the right. Add zeros as placeholders if needed. 2. If 'n' is negative, move the decimal point 'n' places to the left. Add zeros as placeholders if needed.
To order numbers in standard form: 1. First, compare the powers of 10 (the 'n' values). The number with the larger positive power of 10 is greater. The number with the smaller negative power of 10 (i.e., further from zero) is smaller. 2. If the powers of 10 are the same, then compare the 'A' values. The number with the larger 'A' value is greater.
To multiply numbers in standard form: (A × 10ⁿ) × (B × 10ᵐ) = (A × B) × 10ⁿ⁺ᵐ. 1. Multiply the 'A' values together. 2. Add the powers of 10 together. 3. Ensure the resulting 'A' value is between 1 and 10. If not, adjust it and the power of 10 accordingly.
To divide numbers in standard form: (A × 10ⁿ) ÷ (B × 10ᵐ) = (A ÷ B) × 10ⁿ⁻ᵐ. 1. Divide the 'A' values. 2. Subtract the powers of 10 (power of the numerator minus power of the denominator). 3. Ensure the resulting 'A' value is between 1 and 10. If not, adjust it and the power of 10 accordingly.
To add or subtract numbers in standard form, the powers of 10 must be the same. If they are not, adjust one (or both) numbers so that their powers of 10 match. It's often easiest to convert one number so that its power of 10 matches the other, typically choosing the larger power to avoid negative exponents in the 'A' value. 1. Adjust one of the numbers so that both numbers have the same power of 10. 2. Add or subtract the 'A' values. 3. Keep the common power of 10. 4. Ensure the resulting 'A' value is between 1 and 10. If not, adjust it and the power of 10 accordingly.
Key facts to remember
- 1Standard form is written as A × 10ⁿ.
- 2The value of 'A' must be between 1 and 10 (1 ≤ A < 10).
- 3'n' must be an integer (a whole number).
- 4A positive 'n' indicates a large number (decimal point moved left).
- 5A negative 'n' indicates a small number (decimal point moved right).
- 6When multiplying, multiply 'A' values and add powers of 10.
- 7When dividing, divide 'A' values and subtract powers of 10.
- 8For addition or subtraction, ensure the powers of 10 are the same before adding/subtracting the 'A' values.
Worked examples
Example 1
a) Write 73,400,000 in standard form. b) Write 0.0000000405 in standard form. c) Write 6.18 × 10⁵ as an ordinary number. d) Write 2.7 × 10⁻³ as an ordinary number.
Answer
a) 7.34 × 10⁷ b) 4.05 × 10⁻⁸ c) 618,000 d) 0.0027
Example 2
Order the following numbers from smallest to largest: 5.2 × 10⁴, 9.1 × 10³, 3.8 × 10⁵, 7.0 × 10⁴.
Answer
9.1 × 10³, 5.2 × 10⁴, 7.0 × 10⁴, 3.8 × 10⁵
Example 3
Calculate the following, giving your answers in standard form: a) (3 × 10⁶) × (5 × 10⁻²) b) (8.4 × 10⁷) ÷ (2 × 10³) c) (4.5 × 10⁵) + (2.3 × 10⁴)
Answer
a) 1.5 × 10⁵ b) 4.2 × 10⁴ c) 4.73 × 10⁵
For addition/subtraction, it's often easier to adjust the number with the smaller power of 10 to match the larger power, to avoid dealing with 'A' values less than 1.
Common mistakes
- ✗Incorrectly placing the decimal point for 'A' (e.g., writing 12.5 × 10³ instead of 1.25 × 10⁴).
- ✗Getting the sign of 'n' wrong (e.g., writing 0.003 as 3 × 10³ instead of 3 × 10⁻³).
- ✗Errors in counting the number of decimal places moved, leading to an incorrect 'n' value.
- ✗Forgetting to adjust the 'A' value and 'n' after multiplication or division if 'A' falls outside the 1 ≤ A < 10 range.
- ✗Attempting to add or subtract numbers in standard form without first making their powers of 10 the same.
Exam tips
- ★Always double-check that your final answer in standard form has an 'A' value between 1 and 10.
- ★For calculations, show your working step-by-step, even if using a calculator, to gain method marks.
- ★Be meticulous when counting decimal places for 'n', especially with very small numbers.
- ★When ordering numbers, compare the powers of 10 first, as this is usually the quickest way to determine relative size.
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